The rôle of Coulomb branches in 2D gauge theory

Abstract

I give a simple construction of the Coulomb branches ${\mathcal{C}}_{3,4}(G;E)$ of gauge theory in three and four dimensions, defined by H. Nakajima [Adv. Theor. Math. Phys. 20 (2016)] and A. Braverman, M. Finkelberg and H. Nakajima [Adv. Theor. Math. Phys. 22 (2018)] for a compact Lie group $G$ and a polarizable quaternionic representation $E$. The manifolds $\mathcal{C}(G;0)$ are abelian group schemes over the bases of regular adjoint $G_{\mathbb{C}}$-orbits, respectively conjugacy classes, and $\mathcal{C}(G;E)$ is glued together over the base from two copies of $\mathcal{C}(G;0)$ shifted by a rational Lagrangian section ${\epsilon }_V$, representing the Euler class of the index bundle of a polarization $V$ of $E$. Extending the interpretation of ${\mathcal{C}}_3(G;0)$ as “classifying space” for topological 2D gauge theories, I characterize functions on ${\mathcal{C}}_3(G;E)$ as operators on the equivariant quantum cohomologies of $M\times V$, for compact symplectic $G$-manifolds $M$. The non-commutative version has a similar description in terms of the $\Gamma$-class of $V$.